1549380323-Statistical Mechanics Theory and Molecular Simulation

272 Grand canonical ensemble

1

eβPV

∑∞

N=0

eβμN

1

N!h^3 N

∫

dx e−βH(x,N)= 1. (6.4.17)

Taking the exp(βPV) factor to the right side, we obtain

∑∞

N=0

eβμN

1

N!h^3 N

∫

dx e−βH(x,N)= eβPV. (6.4.18)

However, recall that−PV =A ̃(μ,V,T) is the free energy of the grand canonical
ensemble. Thus, exp(βPV) = exp[−β(−PV)] is equal to the partition function. In
the grand canonical ensemble, we denote the partition function asZ(μ,V,T), and it
is given by

Z(μ,V,T) =

∑∞

N=0

eβμN

1

N!h^3 N

∫

dx e−βH(x,N)

=

∑∞

N=0

eβμNQ(N,V,T). (6.4.19)

The productPVis thus related toZ(μ,V,T) by

PV

kT

= lnZ(μ,V,T). (6.4.20)

According to eqn. (6.4.20), the equation of state can be obtained directly from the
partition function in the grand canonical ensemble. Recall, however, that the equation
of state is of the general form (cf. eqn. (2.2.1))

g(〈N〉,P,V,T) = 0, (6.4.21)

which is a function of〈N〉rather thanμ. This suggests that a second equation for the
average particle number〈N〉is needed. By definition,

〈N〉=

1

Z(μ,V,T)

∑∞

N=0

NeβμNQ(N,V,T), (6.4.22)

which can be expressed as a derivative ofZwith respect toμas

〈N〉=kT

(

∂

∂μ

lnZ(μ,V,T)

)

V,T

. (6.4.23)

Eqns. (6.4.23) and (6.4.20) give a prescription for finding the equation of state in the
grand canonical ensemble. Eqn. (6.4.23) must be solved forμin terms of〈N〉and then
substituted back into eqn. (6.4.20) in order to obtain an equation in the proper form.